Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

On the chain rule and change of variables of integrals

Originally published at 狗和留美者不得入内. You can comment here or there.

Theorem 1 (Chain rule) Let [公式] , [公式] , where [公式] and [公式] are open in [公式] , such that [公式] are differentiable on their respective domains. Then [公式] is also differentiable on [公式] , with [公式] for all [公式] .

Proof: We first assume that there exists a neighborhood [公式] of [公式] for which [公式] . This happens in the case of [公式] by inverse function theorem. In that case, by the definition of derivative and its properties, we have

[公式] In the case of [公式] , we have that for all [公式] ,

[公式]

From this, we easily verifies that [公式] , which means that [公式] is differentiable at [公式] and in the case of [公式] , [公式] must hold as well. [公式]

Lemma 1 Let [公式] , [公式] be differentiable [公式] with [公式] and [公式] . Then,

[公式]

Instead of [公式] , one can also use any closed interval of [公式] .

Proof: Follows directly from Fundamental Theorem of Calculus. See Theorem 2 (Newton-Leibniz axiom) of [1]. [公式]

Lemma 1 is a statement of invariance of integral along parameterized smooth paths with the same endpoints.

Theorem 2 (Change of variables or u-substitution in integration) Let [公式] be any differentiable function of [公式] on [公式] , which is continuous on [公式] , and [公式] be Riemann integrable on intervals in its domain. Then,

[公式]

Proof: Let [公式] be an antiderivative of [公式] . By the Fundamental Theorem of Calculus, it suffices to show that the left hand side of [公式] is equal to [公式] , which can be done by applying Lemma 1 accordingly. [公式]

Theorem 3 (Integration by parts) Let [公式] be differentiable functions on [公式] and continuous on [公式] . Then,

[公式]

Proof: We have

[公式]

Rearranging the above completes the proof. [公式]

References

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